Scientific & Engineering Notation Digital Electronics TM 1.1 Foundations and The Board Game Counter 2013 Final Exam Study Guide Project Lead The Way, Inc. Copyright 2009

What is a resistor? What are capacitors? What is the difference from an analog and digital signal? What is a truth table? Properly construct a truth table. Write a Sum-Of-Products (SOP) logic expression from a truth table. Create a truth table given a SOP logic expression. Mathematically express the relationship between the number of input (N) and the number of input combinations. Find simplified SOP logic expression. What is a don’t care condition? What is a Nand gate? What is a NOR gate?

Scientific & Engineering Notation Digital Electronics TM 1.1 Foundations and The Board Game Counter Scientific & Engineering Notation This presentation will demonstrate… How to express numbers in scientific notation. How to express numbers in engineering notation. How to express numbers in SI prefix notation. Project Lead The Way, Inc. Copyright 2009

Scientific & Engineering Notation Digital Electronics TM 1.1 Foundations and The Board Game Counter Scientific Notation Scientific notation is a way of writing very large and very small numbers in a compact form. A number written in scientific notation is written in the form: a × 10b Where: a is a number greater than 1 and less than 9.99 b is an integer Examples: 3.24 × 105 1.435 × 10-7 3.29× 106 7.3 × 10−2 Project Lead The Way, Inc. Copyright 2009

Scientific Notation: Example #1 Scientific & Engineering Notation Digital Electronics TM 1.1 Foundations and The Board Game Counter Scientific Notation: Example #1 Example: Express 5630 in scientific notation. 5 Project Lead The Way, Inc. Copyright 2009

Scientific Notation: Example #2 Scientific & Engineering Notation Digital Electronics TM 1.1 Foundations and The Board Game Counter Scientific Notation: Example #2 Example: Express 0.000628 in scientific notation. 6 Project Lead The Way, Inc. Copyright 2009

Scientific & Engineering Notation Digital Electronics TM 1.1 Foundations and The Board Game Counter Engineering Notation Engineering notation is similar to scientific notation. In engineering notation the powers of ten are always multiples of 3. A number written in engineering notation is written in the form: a × 10b Where: a is a number greater than 1 and less than 999 b is an integer multiple of three Examples: 71.24 × 103 4.32 × 10-6 320.49× 109 123.452 × 10−12 Project Lead The Way, Inc. Copyright 2009

Writing A Number in Engineering Notation Scientific & Engineering Notation Digital Electronics TM 1.1 Foundations and The Board Game Counter Writing A Number in Engineering Notation Shift the decimal point in “groups of three” until the number before the decimal point is between 0 and 999. Multiply by a power of 10 that is equal to the number of places the decimal point has been moved. The power of 10 is positive if the decimal point is moved to the left and negative if the decimal point is moved to the right. Project Lead The Way, Inc. Copyright 2009

Engineering Notation: Example #1 Scientific & Engineering Notation Digital Electronics TM 1.1 Foundations and The Board Game Counter Engineering Notation: Example #1 Example: Express 16346000000 in engineering notation. 9 Project Lead The Way, Inc. Copyright 2009

Engineering Notation: Example #2 Scientific & Engineering Notation Digital Electronics TM 1.1 Foundations and The Board Game Counter Engineering Notation: Example #2 Example: Express 0.0003486 in engineering notation. 10 Project Lead The Way, Inc. Copyright 2009

Scientific & Engineering Notation Digital Electronics TM 1.1 Foundations and The Board Game Counter SI Prefixes SI prefixes are a shorthand way of writing engineering notation for SI numbers. The International System of Units (abbreviated SI from the French Système International d'Unités) is the modern form of the metric system. It is the world's most widely used system of units for science and engineering. Project Lead The Way, Inc. Copyright 2009

Commonly Used SI Prefixes Scientific & Engineering Notation Digital Electronics TM 1.1 Foundations and The Board Game Counter Commonly Used SI Prefixes Value Prefix Symbol 1012 tera T 109 giga G 106 mega M 103 kilo k 10-3 milli m 10-6 micro  10-9 nano n 10-12 pico p 10-15 femto f Project Lead The Way, Inc. Copyright 2009

Scientific & Engineering Notation Digital Electronics TM 1.1 Foundations and The Board Game Counter SI Notation: Example #1 Example: Express 27500  using standard SI notation. (Note:  is the Greek letter omega. In electronics, it is the symbol used for resistance.) 13 Project Lead The Way, Inc. Copyright 2009

Scientific & Engineering Notation Digital Electronics TM 1.1 Foundations and The Board Game Counter SI Notation: Example #2 Example: Express 0.000568 Volts using standard SI notation. 14 Project Lead The Way, Inc. Copyright 2009

Common Electronic Symbol & Units Scientific & Engineering Notation Digital Electronics TM 1.1 Foundations and The Board Game Counter Common Electronic Symbol & Units Quantity Symbol Unit Current I Ampere (A) Voltage V Volt (V) Resistance R Ohm () Frequency f Hertz (Hz) Capacitance C Farad (F) Inductance L Henry (H) Power P Watt (W) Project Lead The Way, Inc. Copyright 2009

Component Identification Digital Electronics TM 1.1 Foundations and The Board Game Counter Resistors A resistor is an electronic component that resists the flow of electrical current. A resistor is typically used to control the amount of current that is flowing in a circuit. Resistance is measured in units of ohms () and named after George Ohm, whose law (Ohm’s Law) defines the fundamental relationship between voltage, current, and resistance. Definition of a Resistor. http://www.wisegeek.com/ Project Lead The Way, Inc. Copyright 2009

How To Read A Resistor’s Value Component Identification Digital Electronics TM 1.1 Foundations and The Board Game Counter How To Read A Resistor’s Value Resistor Color Code Resistor color code chart. Project Lead The Way, Inc. Copyright 2009

Resistor Value: Example #1 Component Identification Digital Electronics TM 1.1 Foundations and The Board Game Counter Resistor Value: Example #1 Example: Determine the nominal value for the resistor shown. Pause the presentation and allow students to complete the example. The solution is on the next slide. Project Lead The Way, Inc. Copyright 2009

Resistor Value: Example #1 Component Identification Digital Electronics TM 1.1 Foundations and The Board Game Counter Resistor Value: Example #1 Example: Determine the nominal value for the resistor shown. Solution: 10 x 100  5% 1000  5% 1 K  5% This slide includes the solution to Resistor Value Example #1. If you print handouts, do not print this page. Project Lead The Way, Inc. Copyright 2009

Resistor Value: Example #3 Component Identification Digital Electronics TM 1.1 Foundations and The Board Game Counter Resistor Value: Example #3 Example: Determine the color bands for a 1.5 K  5% resistor. ? ? ? ? Pause the presentation and allow students to complete the example. The solution is on the next slide. Project Lead The Way, Inc. Copyright 2009

Resistor Value: Example #3 Component Identification Digital Electronics TM 1.1 Foundations and The Board Game Counter Resistor Value: Example #3 Example: Determine the color bands for a 1.5 K  5% resistor. ? ? ? ? Solution: 1.5 K  5% 1500  5% 15 x 100  5% 1: Brown 5: Green 100: Red 5%: Gold This slide includes the solution to Resistor Value Example #3. If you print handouts, do not print this page. Project Lead The Way, Inc. Copyright 2009

Component Identification Digital Electronics TM 1.1 Foundations and The Board Game Counter Capacitors A capacitor is an electronic component that can be used to store an electrical charge. Capacitors are often used in electronic circuits as temporary energy-storage devices. Capacitance is measured in units of farads (F) and named after Michael Faraday, a British chemist and physicist who contributed significantly to the study of electromagnetism. Definition of a Capacitor. Project Lead The Way, Inc. Copyright 2009

How To Read A Capacitor’s Value Component Identification Digital Electronics TM 1.1 Foundations and The Board Game Counter How To Read A Capacitor’s Value Disc Capacitors Code Tolerance A ±0.05% B ±0.1% C ±0.25% D ±0.5% F ±1% G ±2% J ±5% K ±10% M or NONE ±20% N ±30% Q −10%, +30% S −20%, +50% T −10%, +50% Z −20%, +80% 4 7 2 K First Digit First Figure Second Digit Second Figure Third Digit # of Zeros Fourth Digit Tolerance Overview of how a disc capacitor is labeled. 4 7 00 K 4700 pF 10% Note: Units on Disc Capacitors are always in pico-farads Project Lead The Way, Inc. Copyright 2009

Component Identification Digital Electronics TM 1.1 Foundations and The Board Game Counter Capacitor: Example #1 Example: Determine the nominal value for the capacitor shown. Code Tolerance A ±0.05% B ±0.1% C ±0.25% D ±0.5% F ±1% G ±2% J ±5% K ±10% M or NONE ±20% N ±30% Q −10%, +30% S −20%, +50% T −10%, +50% Z −20%, +80% Pause the presentation and allow students to complete the example. The solution is on the next slide. Project Lead The Way, Inc. Copyright 2009

Component Identification Digital Electronics TM 1.1 Foundations and The Board Game Counter Capacitor: Example #1 Example: Determine the nominal value for the capacitor shown. Code Tolerance A ±0.05% B ±0.1% C ±0.25% D ±0.5% F ±1% G ±2% J ±5% K ±10% M or NONE ±20% N ±30% Q −10%, +30% S −20%, +50% T −10%, +50% Z −20%, +80% Solution: 330 pF 5% This slide includes the solution to Capacitor Value Example #1. If you print handouts, do not print this page. Project Lead The Way, Inc. Copyright 2009

Analog and Digital Signals Digital Electronics TM 1.2 Introduction to Analog Analog and Digital Signals Analog Signals Digital Signals Continuous Infinite range of values More exact values, but more difficult to work with Discrete Finite range of values (2) Not as exact as analog, but easier to work with Example: A digital thermostat in a room displays a temperature of 72. An analog thermometer measures the room temperature at 72.482. The analog value is continuous and more accurate, but the digital value is more than adequate for the application and significantly easier to process electronically. This slide defines analog and digital signals and gives several examples of each. Project Lead The Way, Inc. Copyright 2009

Example of Analog Signals Analog and Digital Signals Digital Electronics TM 1.2 Introduction to Analog Example of Analog Signals An analog signal can be any time-varying signal. Minimum and maximum values can be either positive or negative. They can be periodic (repeating) or non-periodic. Sine waves and square waves are two common analog signals. Note that this square wave is not a digital signal because its minimum value is negative. 0 volts Sine Wave Square Wave (not digital) Random-Periodic Examples of common analog signals. Project Lead The Way, Inc. Copyright 2009

Parts of an Analog Signal Analog and Digital Signals Digital Electronics TM 1.2 Introduction to Analog Parts of an Analog Signal Amplitude (peak-to-peak) (peak) Period (T) Frequency: Parts of an analog signal: amplitude, period, & frequency. Project Lead The Way, Inc. Copyright 2009

Example of Digital Signals Analog and Digital Signals Digital Electronics TM 1.2 Introduction to Analog Example of Digital Signals Digital signal are commonly referred to as square waves or clock signals. Their minimum value must be 0 volts, and their maximum value must be 5 volts. They can be periodic (repeating) or non-periodic. The time the signal is high (tH) can vary anywhere from 1% of the period to 99% of the period. 5 volts Examples of common digital signals. 0 volts Project Lead The Way, Inc. Copyright 2009

Parts of a Digital Signal Analog and Digital Signals Digital Electronics TM 1.2 Introduction to Analog Parts of a Digital Signal Amplitude: For digital signals, this will ALWAYS be 5 volts. Period (T): The time it takes for a periodic signal to repeat. (in seconds) Frequency (F): A measure of the number of cycles of the signal per second. (in Hertz, Hz) Time High (tH): The time (in sec.) the signal is high or 5v. Time Low (tL): The time (sec.) the signal is low or 0v. Duty Cycle (DC) (%): The ratio of tH to the period (T), expressed as a percentage. Rising Edge: A 0-to-1 transition of the signal. Falling Edge: A 1-to-0 transition of the signal. Falling Edge Time High (tH) Amplitude Time Low (tL) Rising Edge Period (T) The parts of a digital signal: amplitude, period & frequency, time high, time low, duty cycle, rising & falling edge. Frequency (F): Duty Cycle (%): Project Lead The Way, Inc. Copyright 2009

Example: Digital Signal Analog and Digital Signals Digital Electronics TM 1.2 Introduction to Analog Example: Digital Signal Example: Determine the following information for the digital signal shown: Amplitude Period (T) Frequency (f) Time High (tH) Time Low (tL) Duty Cycle (DC) Pause the presentation and allow the student to work on the example. The solution is on the next two slides. Project Lead The Way, Inc. Copyright 2009

Example: Digital Signal Analog and Digital Signals Digital Electronics TM 1.2 Introduction to Analog Example: Digital Signal Solution: Amplitude: Period (T): Frequency (F): 4V 2V 2ms 4ms Here is the solution. If you print handouts, do not print this page. (1 of 2) Project Lead The Way, Inc. Copyright 2009

Example: Digital Signal Analog and Digital Signals Digital Electronics TM 1.2 Introduction to Analog Example: Digital Signal Solution: Time High (tH): Time Low (tL): Duty Cycle (DC) %: Here is the solution. If you print handouts, don’t print this page. (2 of 2) Project Lead The Way, Inc. Copyright 2009

Decimal ‒to‒ Binary Conversion Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Decimal ‒to‒ Binary Conversion The Process : Successive Division Divide the Decimal Number by 2; the remainder is the LSB of Binary Number . If the quotient is zero, the conversion is complete; else repeat step (a) using the quotient as the Decimal Number. The new remainder is the next most significant bit of the Binary Number. Example: Convert the decimal number 610 into its binary equivalent. Review the DECIMAL-to-BINARY conversion process. Remind the students to subscript all numbers (i.e. Subscript 10 for decimal & subscript 2 for binary) A common mistake is inverting the LSB and MSB. The three-dot triangular symbol here stands for the word “therefore” and is used commonly among mathematics scholars.  610 = 1102 Project Lead The Way, Inc. Copyright 2009

Dec → Binary : Example #1 Example: Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Dec → Binary : Example #1 Example: Convert the decimal number 2610 into its binary equivalent. Pause the power point and allow the student to work on the example. The solution is on the next slide. Project Lead The Way, Inc. Copyright 2009

Dec → Binary : Example #1 Example: Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Dec → Binary : Example #1 Example: Convert the decimal number 2610 into its binary equivalent. Solution:  2610 = 110102 Here is the solution. If you print handouts, don’t print this page. Project Lead The Way, Inc. Copyright 2009

Binary ‒to‒ Decimal Process Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Binary ‒to‒ Decimal Process The Process : Weighted Multiplication Multiply each bit of the Binary Number by it corresponding bit- weighting factor (i.e. Bit-0→20=1; Bit-1→21=2; Bit-2→22=4; etc). Sum up all the products in step (a) to get the Decimal Number. Example: Convert the decimal number 01102 into its decimal equivalent. 1 23 22 21 20 8 4 2 + = 610 Review the BINARY-to-DECIMAL conversion process. Remind the students to subscript all numbers (i.e. Subscript 10 for decimal & subscript 2 for decimal) Let the students know that as the become more proficient at the conversions, they may not need to write out the Bit-Weighting Factors.  0110 2 = 6 10 Bit-Weighting Factors Project Lead The Way, Inc. Copyright 2009

Binary → Dec : Example #1 Example: Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Binary → Dec : Example #1 Example: Convert the binary number 100102 into its decimal equivalent. Pause the power point and allow the student to work on the example. The solution is on the next slide. Project Lead The Way, Inc. Copyright 2009

Base10 Base2 Base10 Base2 Summary & Review DECIMAL BINARY DECIMAL Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Summary & Review Base10 DECIMAL Base2 BINARY Successive Division Divide the Decimal Number by 2; the remainder is the LSB of Binary Number . If the Quotient Zero, the conversion is complete; else repeat step (a) using the Quotient as the Decimal Number. The new remainder is the next most significant bit of the Binary Number. Weighted Multiplication Base10 DECIMAL Base2 BINARY Prior to assigning the activity, review the process for DECIMAL-to-BINARY and BINARY-to-DECIMAL. Multiply each bit of the Binary Number by it corresponding bit-weighting factor (i.e. Bit-0→20=1; Bit-1→21=2; Bit-2→22=4; etc). Sum up all the products in step (a) to get the Decimal Number. Project Lead The Way, Inc. Copyright 2009

Karnaugh Mapping Digital Electronics 2.2 Intro to NAND & NOR Logic K-Map Format Each minterm in a truth table corresponds to a cell in the K-Map. K-Map cells are labeled such that both horizontal and vertical movement differ only by one variable. Since the adjacent cells differ by only one variable, they can be grouped to create simpler terms in the sum-of- products expression. The sum-of-products expression for the logic function can be obtained by OR-ing together the cells or group of cells that contain 1s. Explain how a K-Map is formatted and how a simplified logic expression can be written from the cell groupings. Project Lead The Way, Inc. Copyright 2009

K-Map Simplification Process Karnaugh Mapping Digital Electronics 2.2 Intro to NAND & NOR Logic K-Map Simplification Process Construct a label for the K-Map. Place 1s in cells corresponding to the 1s in the truth table. Place 0s in the other cells. Identify and group all isolated 1’s. Isolated 1’s are ones that cannot be grouped with any other one, or can only be grouped with one other adjacent one. Group any hex. Group any octet, even if it contains some 1s already grouped but not enclosed in a hex. Group any quad, even if it contains some 1s already grouped but not enclosed in a hex or octet. Group any pair, even if it contains some 1s already grouped but not enclosed in a hex, octet, or quad. OR together all terms to generate the SOP equation. Steps for simplifying using the K-Map process. Project Lead The Way, Inc. Copyright 2009

Example #1: 2 Variable K-Map Karnaugh Mapping Digital Electronics 2.2 Intro to NAND & NOR Logic Example #1: 2 Variable K-Map Example: After labeling and transferring the truth table data into the K-Map, write the simplified sum-of-products (SOP) logic expression for the logic function F1. V J K F1 1 Pause the presentation and allow the student to complete the example. The solution is on the next slide. Project Lead The Way, Inc. Copyright 2009

Example #1: 2 Variable K-Map Karnaugh Mapping Digital Electronics 2.2 Intro to NAND & NOR Logic Example #1: 2 Variable K-Map Example: After labeling and transferring the truth table data into the K-Map, write the simplified sum-of-products (SOP) logic expression for the logic function F1. Solution: V 1 J K F1 1 Here is the solution. If you print handouts, do not print this page. Project Lead The Way, Inc. Copyright 2009

Example #2: 3 Variable K-Map Karnaugh Mapping Digital Electronics 2.2 Intro to NAND & NOR Logic Example #2: 3 Variable K-Map Example: After labeling and transferring the truth table data into the K-Map, write the simplified sum-of-products (SOP) logic expression for the logic function F2. E F G F2 1 Pause the presentation and allow students to complete the example. The solution is on the next slide. Project Lead The Way, Inc. Copyright 2009

Example #2: 3 Variable K-Map Karnaugh Mapping Digital Electronics 2.2 Intro to NAND & NOR Logic Example #2: 3 Variable K-Map Example: After labeling and transferring the truth table data into the K-Map, write the simplified sum-of-products (SOP) logic expression for the logic function F2. Solution: V 1 E F G F2 1 Here is the solution. If you print handouts, do not print this page. Project Lead The Way, Inc. Copyright 2009

Truth Table to K-Map Mapping Karnaugh Mapping Digital Electronics 2.2 Intro to NAND & NOR Logic Truth Table to K-Map Mapping Four Variable K-Map W X Y Z FWXYZ Minterm – 0 Minterm – 1 1 Minterm – 2 Minterm – 3 Minterm – 4 Minterm – 5 Minterm – 6 Minterm – 7 Minterm – 8 Minterm – 9 Minterm – 10 Minterm – 11 Minterm – 12 Minterm – 13 Minterm – 14 Minterm – 15 Only one variable changes for every column change 1 V 1 3 2 4 5 7 6 12 13 15 14 8 9 11 10 1 Only one variable changes for every row change Demonstrate how the minterms of a four variable truth table are mapped to a K-Map. 1 1 Project Lead The Way, Inc. Copyright 2009

Karnaugh Mapping Digital Electronics 2.2 Intro to NAND & NOR Logic Don’t Care Conditions A don’t care condition, marked by (X) in the truth table, indicates a condition where the design doesn’t care if the output is a (0) or a (1). A don’t care condition can be treated as a (0) or a (1) in a K-Map. Treating a don’t care as a (0) means that you do not need to group it. Treating a don’t care as a (1) allows you to make a grouping larger, resulting in a simpler term in the SOP equation. Project Lead The Way, Inc. Copyright 2009

Some You Group, Some You Don’t Karnaugh Mapping Digital Electronics 2.2 Intro to NAND & NOR Logic Some You Group, Some You Don’t This don’t care condition was treated as a (1). This allowed the grouping of a single one to become a grouping of two, resulting in a simpler term. V X 1 Explain that you include don’t care conditions only if it allows you to make a grouping larger. There was no advantage in treating this don’t care condition as a (1), thus it was treated as a (0) and not grouped. Project Lead The Way, Inc. Copyright 2009

Universal Gate – NAND This presentation will demonstrate Digital Electronics 2.2 Intro to NAND & NOR Logic Universal Gate – NAND This presentation will demonstrate The basic function of the NAND gate. How a NAND gate can be used to replace an AND gate, an OR gate, or an INVERTER gate. How a logic circuit implemented with AOI logic gates can be re-implemented using only NAND gates. That using a single gate type, in this case NAND, will reduce the number of integrated circuits (IC) required to implement a logic circuit. Introductory Slide / Overview of Presentation AOI Logic NAND Logic More ICs = More $$ Less ICs = Less $$ Project Lead The Way, Inc Copyright 2009

Universal Gate - NAND Digital Electronics 2.2 Intro to NAND & NOR Logic NAND Gate X Y X Y Z 1 Overview of basic NAND gate : Logic Symbol, Logic Expression (using DeMorgan’s) and Truth Table. Project Lead The Way, Inc Copyright 2009

NAND Gate as an Inverter Gate Universal Gate - NAND Digital Electronics 2.2 Intro to NAND & NOR Logic NAND Gate as an Inverter Gate (Before Bubble) X X Z 1 When you tie the inputs on a NAND gate together, the output will be the complement of the input. Equivalent to Inverter Project Lead The Way, Inc Copyright 2009

NAND Gate as an AND Gate X Y X Y Z 1 NAND Gate Inverter Universal Gate - NAND Digital Electronics 2.2 Intro to NAND & NOR Logic NAND Gate as an AND Gate X Y NAND Gate Inverter X Y Z 1 This one is easy to see, a NAND gate is an AND gate with the output inverted. So if you invert the output again, you will get an AND gate. Note that we are using a NAND gate for the inverter. Equivalent to AND Gate Project Lead The Way, Inc Copyright 2009

NAND Gate as an OR Gate X Y X Y Z 1 NAND Gate Inverters Universal Gate - NAND Digital Electronics 2.2 Intro to NAND & NOR Logic NAND Gate as an OR Gate X Y NAND Gate Inverters X Y Z 1 This one is a bit harder to see. If you invert both of the inputs of a NAND gate, you will get an OR gate. Note that we’re using NAND gates as inverters. Equivalent to OR Gate Project Lead The Way, Inc Copyright 2009

NAND Gate Equivalent to AOI Gates Universal Gate - NAND Digital Electronics 2.2 Intro to NAND & NOR Logic NAND Gate Equivalent to AOI Gates AND OR INVERTER Summary of the three AOI gates and their NAND equivalent. Project Lead The Way, Inc Copyright 2009

Universal Gate – NOR This presentation will demonstrate… Digital Electronics 2.2 Intro to NAND & NOR Logic Universal Gate – NOR This presentation will demonstrate… The basic function of the NOR gate. How an NOR gate can be using to replace an AND gate, an OR gate or an INVERTER gate. How a logic circuit implemented with AOI logic gates could be re-implemented using only NOR gates That using a single gate type, in this case NOR, will reduce the number of integrated circuits (IC) required to implement a logic circuit. Introductory Slide / Overview of Presentation AOI Logic NOR Logic More ICs = More $$ Less ICs = Less $$ Project Lead The Way, Inc. Copyright 2009

Universal Gate - NOR Digital Electronics 2.2 Intro to NAND & NOR Logic NOR Gate X Y X Y Z 1 Overview of basic NOR gate : Logic Symbol, Logic Expression (using DeMorgan’s) and Truth Table. Project Lead The Way, Inc. Copyright 2009

NOR Gate as an Inverter Gate Universal Gate - NOR Digital Electronics 2.2 Intro to NAND & NOR Logic NOR Gate as an Inverter Gate X (Before Bubble) X Z 1 When you tie the inputs on a NOR gate together, the output will be the complement of the input. Equivalent to Inverter Project Lead The Way, Inc. Copyright 2009

NOR Gate as an OR Gate X Y X Y Z 1 NOR Gate “Inverter” Universal Gate - NOR Digital Electronics 2.2 Intro to NAND & NOR Logic NOR Gate as an OR Gate X Y NOR Gate “Inverter” X Y Z 1 This one is easy to see, a NOR gate is in NOR gate with the output inverted. So if you invert the output again you will get an OR gate. Note we’re using a NOR gate for the inverter. Equivalent to OR Gate Project Lead The Way, Inc. Copyright 2009

NOR Gate as an AND Gate X Y X Y Z 1 NOR Gate “Inverters” Universal Gate - NOR Digital Electronics 2.2 Intro to NAND & NOR Logic NOR Gate as an AND Gate X Y “Inverters” NOR Gate X Y Z 1 This one is a bit harder to see. If you invert both the inputs of a NOR gate you will get an AND gate. Note we’re using NOR gates as inverters. Equivalent to AND Gate Project Lead The Way, Inc. Copyright 2009

NOR Gate Equivalent of AOI Gates Universal Gate - NOR Digital Electronics 2.2 Intro to NAND & NOR Logic NOR Gate Equivalent of AOI Gates AND OR INVERTER Summary of the three AOI gates and their NOR equivalent. Project Lead The Way, Inc. Copyright 2009

AOI Logic Implementation Digital Electronics 2.1 Introduction to AOI Logic AOI Logic Implementation This presentation will demonstrate how to… Design an AOI logic circuit from a Sum-Of- Products (SOP) logic expression. Introductory Slide / Overview of Presentation EQUALS AOI Logic Circuit Logic Expression Project Lead The Way, Inc. Copyright 2009

Designing AOI SOP Logic Circuits AOI Logic Implementation Digital Electronics 2.1 Introduction to AOI Logic Designing AOI SOP Logic Circuits Design Steps Implement each Minterm in the logic expression with an AND gate with the same number of inputs as there are variables in the Minterm. (i.e., AB = 2 input gate, ABC = 3 input gate, ABCD = 4 input gate, etc.) OR together the outputs of the AND gates to produce the logic expression. If necessary, gates can be cascaded to create gates with more inputs. Design Steps for SOP AOI logic designs. Project Lead The Way, Inc. Copyright 2009

Example #1: AOI Implementation SOP AOI Logic Implementation Digital Electronics 2.1 Introduction to AOI Logic Example #1: AOI Implementation SOP Example Design an AOI Logic Circuit for the SOP logic expression shown below. Pause the power point and allow the student to work on the example. The solution is on the next slide. Project Lead The Way, Inc. Copyright 2009

Example #2: AOI Implementation SOP AOI Logic Implementation Digital Electronics 2.1 Introduction to AOI Logic Example #2: AOI Implementation SOP Example Unfortunately, in this class, we only have access to (2) input OR gates and (2) & (3) input AND gates. Limiting your design to these gates, redesign the AOI Logic Circuit for the SOP expression in the previous example. Pause the power point and allow the student to work on the example. The solution is on the next slide. Project Lead The Way, Inc. Copyright 2009

Example #2: AOI Implementation SOP AOI Logic Implementation Digital Electronics 2.1 Introduction to AOI Logic Example #2: AOI Implementation SOP Example Unfortunately, only (2) input OR gates and (2) & (3) AND gates exist in hardware. Limiting your design to these gates, redesign the AOI Logic Circuit for the SOP expression in the previous example. Solution The solution for example #2 is included on this slide. If you print handouts, do not print this page. Project Lead The Way, Inc. Copyright 2009